Hankel loop integral

SearchAdvancedHelp

(0.001 seconds)

7 matching pages

1: 5.9 Integral Representations

… ►

Hankel’s Loop Integral

… ►

►►

Figure 5.9.1: $t$-plane. Contour for Hankel’s loop integral. Magnify

…

2: 10.9 Integral Representations

… ►

Mehler–Sonine and Related Integrals

… ►

Schläfli–Sommerfeld Integrals

… ►

Hankel’s Integrals

►In (10.9.20) and (10.9.21) the integration paths are simple loop contours not enclosing $t=-1$. … ►

§10.9(iv) Compendia

…

3: 25.11 Hurwitz Zeta Function

… ►

§25.11(vii) Integral Representations

… ►

25.11.30 $$\zeta (s,a)=\frac{\mathrm{\Gamma }\left(1-s\right)}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}\frac{{\mathrm{e}}^{az}{z}^{s-1}}{1-{\mathrm{e}}^z}dz,$$ $s\ne 1$, $\mathrm{\Re }a>0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

contour integral, integral representation

Source:

Erdélyi et al. (1953a, (1.10.5), p. 25); with $z=-t$

Proof sketch:

This contour integral uses Hankel’s contour Figure 5.9.1. Assume $\mathrm{\Re }s>1$, collapse the integration path onto the negative real axis, apply (25.11.25), followed by analytic continuation for all $s\ne 1$, $\mathrm{\Re }a>0$, since the integrand is analytic in $z$, the convergence at the ends of the path is exponential for all $s\ne -1$ and the Hankel contour can be kept well clear of the singularity at the origin in the $z$-plane.

Referenced by:

§25.11(i), §25.11(vii), Erratum (V1.2.1) for Figure 5.9.1

Permalink:

http://dlmf.nist.gov/25.11.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vii), §25.11 and Ch.25

►where the integration contour (see Figure 5.9.1) is a loop around the negative real axis as described for (25.5.20). ►

§25.11(viii) Further Integral Representations

… ►

§25.11(ix) Integrals

…

4: 13.23 Integrals

§13.23 Integrals

… ►For the particular loop contour, see Figure 5.9.1. … ►

§13.23(iii) Hankel Transforms

… ►For additional Hankel transforms and also other Bessel transforms see Erdélyi et al. (1954b, §8.18) and Oberhettinger (1972, §1.16 and 3.4.42–46, 4.4.45–47, 5.94–97). …

5: 13.10 Integrals

… ►

Loop Integrals

… ►For the particular loop contour, see Figure 5.9.1. … ►

§13.10(v) Hankel Transforms

… ►For additional Hankel transforms and also other Bessel transforms see Erdélyi et al. (1954b, §8.18) and Oberhettinger (1972, §§1.16 and 3.4.42–46, 4.4.45–47, 5.94–97). …

6: Bibliography C

… ►

Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams. J. Comput. Appl. Math. 115 (1-2), pp. 93–99.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt

Referenced by:

§16.24(ii).

Encodings:

BibTeX

… ►

Determination of $\nu$-zeros of Hankel functions. Comput. Phys. Comm. 32 (3), pp. 333–339.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 7S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

… ►

A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

ⓘ

External Links:

ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

… ►

Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.

ⓘ

External Links:

Document

Referenced by:

§10.74(vii).

Encodings:

BibTeX

… ►

Computation of Hankel transforms. SIAM Rev. 14 (2), pp. 278–285.

ⓘ

External Links:

ISSN 1095-7200, Document, MathReview (R. D. Riess), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

…

7: 13.4 Integral Representations

§13.4 Integral Representations

… ►

§13.4(ii) Contour Integrals

… ► … ►

§13.4(iii) Mellin–Barnes Integrals

… ►